Papers

Let $\rho$ and $\sigma$ be density operators on a separable Hilbert space. For $0<\alpha<1$, we study the directed trace-norm overlap $\Phi_\alpha(\rho,\sigma)=\|\rho^\alpha\sigma^{1-\alpha}\|_1$ and its symmetrized form $\mathcal F_\alpha(\rho,\sigma)=\frac12\bigl(\Phi_\alpha(\rho,\sigma)+\Phi_{1-\alpha}(\rho,\sigma)\bigr)$. At $\alpha=\tfrac12$, both quantities coincide with the root Uhlmann–Jozsa fidelity. Our aim is not to introduce a new notion of fidelity, but to understand how these overlaps vary with the parameter and when equality occurs in the resulting inequalities. We first prove that $\alpha\mapsto\Phi_\alpha(\rho,\sigma)$ is log-convex. This yields a sharp lower bound for $\mathcal F_\alpha$ in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that $\Phi_\alpha$ is the trace functional $Q_{\alpha,1/2}$ associated with the $\alpha$-$z$ Renyi divergence, which connects the directed overlap with the known $\alpha$-$z$ Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at $\alpha=\tfrac12$; for every other value of $\alpha$, it already fails under diagonal pinching of faithful real qubit states. In finite dimensions, we give a complete spectral description of the equality cases and provide explicit noncommuting examples. We also discuss some natural questions about overlap-preserving maps. Finally, we prove that the difference between the Petz overlap and the directed trace-norm overlap vanishes exactly when $\rho$ and $\sigma$ commute, without requiring either faithfulness or any additional support assumption.

Let $\mathscr L$ denote the set of Liouville numbers. We prove a local arithmetic rigidity theorem for real-analytic functions. If $U\subset\mathbb R$ is an open interval and $f:U\to\mathbb R$ is real-analytic with $f(\mathscr L\cap U)\subseteq\mathscr L$, then $f$ is the restriction to $U$ of a rational function in $\mathbb R(x)$. Quantitatively, there exists an absolute constant $\tau>2$ such that, for every nonrational real-analytic function $f$ and every nonempty open subinterval $V\subset U$, the set of $\xi\in V\cap\mathscr L$ satisfying $\mu(f(\xi))\leq\tau$ contains a Cantor set. As a consequence, every entire function $F:\mathbb C\to\mathbb C$ satisfying $F(\mathscr L)\subseteq \mathscr L$ is a polynomial with real coefficients. This gives a negative answer to a problem posed by Mahler in 1984. The proof develops a two-height counting estimate for rational approximation to analytic graphs, treating source and target denominators separately, and combines it with a nested construction producing Liouville inputs whose images have uniformly bounded irrationality exponent.

We prove a lower comparison for disorder-averaged ferromagnetic XY correlations: independent nonnegative random couplings $K_e$ can be replaced by $-\frac12\log\mathbb{E}e^{-2K_e}$. Ginibre's inequality makes each normalized correlation convex in $1-e^{-2K_e}$, so conditional Jensen and thinning compare Bernoulli bond densities; two star-wise Jensen passes handle site dilution on bipartite graphs. Together with the phase-existence theorem of Dario and Garban, these comparisons give local Lipschitz regularity of both the averaged algebraic-phase and susceptibility-divergence thresholds throughout the supercritical density intervals on $\mathbb{Z}^2$. The clean finite-domain criterion of van Engelenburg and Lis also yields explicit finite-size susceptibility lower bounds in a weak-dilution region. For bounded centered bond noise, the possible suppression of the algebraic-phase temperature is at most quadratic in the noise amplitude. Finally, at finite positive thresholds we prove total-variation continuity for bounded iid coupling laws whose essential supremum equals the common support cap, with local Lipschitz control when the reference law has an atom at that cap.

We present a procedure that, given a pair of coprime integers, produces a one-vertex triangulation of the three-sphere containing the corresponding torus knot as an edge. The construction is canonical, and produces a three-sphere triangulation from a pair of minimal layered triangulations of the solid torus. The size of the triangulation is, in the worst case, linear, but can be as small as logarithmic in the crossing number of the knot, motivating this triangulation size as an organising principle for knots, alternative to the crossing number.

We develop controlled B-series and controlled Runge-Kutta methods as extensions of classical B-series and Runge-Kutta theory to controlled-driven rough differential equations. The driving signal is a path controlled by an underlying $\alpha$-Hölder step-$2$ rough path, with $\alpha\in(1/3,1/2]$, as arises when the output of one rough system drives another. Controlled B-series retain the classical rooted-tree and elementary-differential structure, while incorporating driver-dependent coefficients defined recursively through controlled rough integration. We establish finite-order expansions of the exact solution and its controlled Runge-Kutta approximation, and derive corresponding tree-based order conditions. Under suitable smoothness, solvability, stability, and boundedness assumptions, matching through tree order $p$ yields local error $O(|t-s|^{(p+1)\alpha})$ and global error $O(h^{(p+1)\alpha-1})$, provided $(p+1)\alpha>1$. The standard rough differential equation formulation is recovered when the controlled driver is the reference path itself, and the classical time-driven case recovers the usual B-series and Runge-Kutta methods. We further construct a simplified method using only increments of the controlled driver. Segmentwise canonical lifting of its piecewise linear interpolation gives explicit tree coefficients and reduces the third-order conditions to the classical ones. Assuming a Wong-Zakai solution-approximation rate $r_0>0$, the simplified method has global error $O(h^{\min\{r_0,(p+1)\alpha-1\}})$. Numerical experiments with fractional Brownian controlled drivers illustrate the resulting convergence behaviour.

We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target $p$ is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of $\omega=\log(p/q)$ yields a coupled parabolic-elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of $p$ by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive $C^{2+\alpha}$ target weights and compatible $C^{2+\alpha}$ initial data. For long-time behavior, the oscillation of $\omega$ decays at the explicit rate $e^{-t}$ and the solution density exponentially converges to target density. Moreover, an exact entropy-dissipation identity holds relative to the target density, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate $e^{-t}$ without a convexity assumption on the target.

Global inversion of nonlinear multivariable systems raises the question of whether every target value of the selected output derivatives can be realized by a plant state. We prove that, under globally regular modified inversion with fixed row selection and finite termination at full input rank, every fixed homogeneous linear output assignment yields a surjective output-derivative map with a smooth global section. The result requires a state bound on each finite interval in terms of the initial state and finitely many output derivatives, uniformly over inputs. This condition is strictly weaker than weak uniform zero-state detectability, making the separate surjectivity requirement in a global inversion approach to stabilization a consequence of detectability under the stated inversion hypotheses. The section selects a realizing state smoothly for every derivative target, including in the presence of internal dynamics, and is constructed through complete lifting of compatible smooth references. For known plants, the same lifting method yields a smooth dynamic controller using full state feedback and current reference derivatives, with initialization from the plant state. From any initial state, it achieves exact tracking of any smooth desired output after any prescribed positive time.

We prove that every finite subset of the integers which tiles by translations satisfies the Coven-Meyerowitz condition T2, with no restriction on the number of prime factors or their exponents. Together with the necessity of T1 and the sufficiency of T1 and T2 proved by Coven and Meyerowitz, this gives their proposed characterization of finite integer tiles. The proof uses strong induction on a cyclic tiling period. Character identities produce periodic Boolean product stripes; integral descent to a coprime quotient and the Frobenius identity force a common orientation. Independent phase shifts then give smaller-period tilings from which the mixed cyclotomic zeros of the original factors can be recovered. A companion Lean formalization verifies the unrestricted T2 necessity statement.

Let $M$ be a complex surface endowed with a holomorphic projective connection and let $C \subset M$ be a compact smooth holomorphic curve. Given two singular holomorphic foliations $\mathcal{F}$ and $\mathcal{G}$ near $C$, with $C$ not $\mathcal{G}$-invariant, we attach to every point $p \in C$ an index $\mathrm{Ind}(\mathcal{F},\mathcal{G},C,p) \in \mathbb{C}$ of $\mathcal{F}$ relative to the reference $\mathcal{G}$, and we prove that these indices add up to $T\mathcal{G} \cdot C$. When $C$ is $\mathcal{F}$-invariant the index is the Camacho–Sad index corrected by the order of tangency of $\mathcal{G}$ with $C$, and the index formula reduces to the Camacho–Sad formula. The main applications are two contact formulas. At a tangency point $p$ of $\mathcal{G}$ with $C$, let $k(\mathcal{G},C,p)$ be the ratio of the curvatures at $p$ of the leaf of $\mathcal{G}$ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If $\mathcal{G}$ has no singular points on $C$ and only simple tangencies with $C$, then $$ \sum_{p} \frac{1}{1 - k(\mathcal{G},C,p)} = \frac{2}{3}\left( C \cdot C + g - 1 \right), $$ where $g$ is the genus of $C$: the number of tangencies depends on $\mathcal{G}$, but this weighted count does not. For a pencil of lines in the projective plane it is the Plücker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to $\mathcal{F}$ and $\mathcal{G}$ which is not a geodesic, the total index of the tangencies of $\mathcal{F}$ with $\mathcal{G}$ along $C$ equals the number of tangencies of $\mathcal{G}$ with $C$ minus one third of the number of inflection points of $C$; in particular, it does not depend on $\mathcal{F}$.

For every integer $n\ge2$, we construct a smooth complete Riemannian metric $G_n$ on $\mathbb{R}^n$ with full curvature norm at most one and injectivity radius at least one for which no $C^2$ isometric immersion into a finite-dimensional Euclidean space has bounded mean curvature. In dimension two, bounded second fundamental form would give uniformly controlled finite Jacobian representations of the Laplacian of the conformal factor. We construct disjoint conformal blocks for which the curvature remains bounded while the finite Jacobian representation cost tends to infinity. Taking Euclidean products gives all higher dimensions. The result answers Yau's Problem~52 negatively under the stronger assumption of bounded full curvature.

We address two questions posed by Qu and Valkó in their study of the pair correlation function of the $\mathrm{Sine}_\beta$ process. First, we prove that the pair correlation function admits a jointly continuous version in the inverse-temperature and spatial parameters on $(0,\infty)\times\mathbb{R}$, removing the restriction $\beta>2$ in their joint-continuity result. The argument uses a general observation: separate weak continuity of a family of probability laws, together with stochastic monotonicity in one parameter, implies joint weak continuity. Applied to the terminal value of the Qu–Valkó diffusion, their Palm-density formula then gives the result without differentiating the Fourier expansion. Second, for $\beta=2n$ we give a direct proof that the Qu–Valkó matrix-recursion power series agrees with the Selberg-integral representation recorded by Forrester. A Krawtchouk transform converts the Qu–Valkó system into a $(2n+1)$-dimensional differential system with parameter $-(n+1)$, while a centered Aomoto–Selberg trace system has parameter $n+1$. An explicit triangular differential intertwiner reflects the parameter $\eta\mapsto-\eta$. Matching the unique Frobenius branch of exponent $2$ gives exactly the Selberg normalization. As a consequence, every coefficient of the Qu–Valkó recursion is identified with an even centered moment of the corresponding Jacobi–Selberg trace statistic.

We give eight-vertex counterexamples to the equality characterization in Conjecture 20 of Chen, Guo, Li and Wang, Electron. J. Combin. 32(4) (2025), P4.37. The examples occur at the included endpoint $k=3n/4$ and extend to infinite families, including a family for which both the graph and its complement are connected. These examples do not contradict the numerical inequality in that conjecture.

Let N be a sufficiently large odd integer. We study the representation of a large integer m N^k as a sum of m almost equal k-th powers of primes, where the common distribution scale is at least N^theta. Assuming the Generalized Riemann Hypothesis (GRH), we prove that for any epsilon > 0 one may take theta > 1/2, provided that m < k(k+1)/2 and that m, k are odd integers sufficiently large. This is achieved by employing a new tool closely related to earlier work. For sufficiently large m, by refining the discussion of the minor arcs, we further show that under GRH the equation has a solution provided m > 16k log k + 4k + 2. Moreover, we give an asymptotic formula for the number of solutions, with main term involving N^{2k theta (1 - (1 - 1/k)^{m_1}) + theta(m-1) + 1 - k}, where m_1 = 4k. Under GRH and for sufficiently large m, k, the short-interval Waring-Goldbach problem is essentially settled.

Single-machine asymmetric weighted earliness–tardiness (AWET) scheduling asks how to sequence jobs around a common synchronization date when early and late completion incur unrelated job-dependent penalties. At the boundary nonrestrictive date $d=\sum_jp_j$, a compact V-shaped schedule reduces the continuous-time problem to a quadratic choice of a nonempty early set. We establish four complementary results for this model. First, the positive-integer problem is strongly NP-complete by a unary-polynomial reduction from Restricted Exact Cover by 3-Sets. Second, unrestricted AWET admits a polynomial-time $(3+2\sqrt2+\varepsilon)$-approximation based on an anchored semidefinite relaxation and deterministic marginal thresholding. Third, when the earliness and tardiness ratio orders are strict reversals, the problem is weakly NP-complete but has an exact two-resource pseudopolynomial dynamic program. Fourth, for fixed total refinements whose ratio permutation is separable, an exact separating-tree recurrence and coordinated geometric trimming yield an FPTAS. The proofs use different manifestations of the same canonical objective: scale-separated prefix penalties, positive-semidefinite minimum-kernel covariance, a dominant completed load square, and a bounded four-coordinate decomposition interface. Together, the results show that the decisive issue is not merely whether the two ratio orders agree, but whether their interaction can be controlled by a global certificate or compressed into a bounded constructive interface.

We give a polynomial-size linear programming bound for $q$-ary all-symbol locally recoverable codes with locality parameters $(r,\delta)$, without assuming linearity. The key idea is to keep, for every ordered pair of codewords and every selected recovery view, the joint Hamming weight on the helper set, the recovered coordinate, and the rest of the code – rather than collapsing this triple into a single distance, as earlier formulations do. Averaging this three-block distribution over recovery views of the same length yields exact identities linking it to the global distance distribution, together with nonnegative product-Krawtchouk constraints that encode locality and spectral positivity simultaneously. The resulting LP has polynomially many variables, its optimum dominates the ordinary Delsarte bound, and an earlier outside-distance formulation, the convex-hull bound of Li–Wei–Xiong, and the dual-based bound of Gruica–Jany–Ravagnani all arise from it as coarser marginals. Exact rational certificates over $q=2,3,4$ show the bound is strictly stronger than the best of these prior LPs in thirteen of fifteen tested cases, pinning down seven exact maximum code sizes and twelve exact maximum linear dimensions.

We propose a new class of sigma models that extend AKSZ sigma models, which we refer to as gauged AKSZ sigma models. These models are constructed using QP-manifolds. The compatibility between the AKSZ structure and gauging is characterized by geometric conditions on the target graded manifold. As an example, we explicitly analyze the gauging of higher Courant sigma models in higher dimensions. When fluxes and boundaries are introduced into gauged higher Courant sigma models, generalizations of momentum maps and Hamiltonian $G$-spaces arise as consistency conditions for the models.

We construct a smooth, strictly convex ring that admit a unique solution of Gaussian $p$-Capacity problem whose super-level sets are nonconvex. This result provide a negative answer.

Extremum seeking control (ESC) relies on deliberately injected excitation to extract optimization information from measured outputs, making its networked implementation particularly vulnerable to denial-of-service (DoS) attacks. This paper develops a resilient discrete-time ESC architecture for cyber-physical systems subject to DoS attacks. By holding the most recently successfully transmitted signals during DoS intervals, the proposed mechanism yields averaged error dynamics with an exponentially contracting mode under successful communication and a neutral mode under attack. For deterministic DoS attacks, practical exponential convergence to the extremum is established under an average bound on the attack duration, with an explicit convergence rate depending on the fraction of time under attack. For probabilistic DoS attacks, almost-sure and in-probability convergence properties are established via stochastic averaging, with the attack success probability explicitly entering the convergence rate. A key finding is that a seemingly natural zero-input strategy can fundamentally compromise ESC: attacks synchronized with the dither generate an $\mathcal{O}(1/a)$ bias in the averaged dynamics, where $a$ is the dither amplitude, and can displace the equilibrium from the true optimizer. Thus, under the proposed hold-based architecture, increasingly severe DoS attacks primarily slow the optimization process rather than destroy its stability, provided that communication is not permanently blocked. Numerical simulations illustrate the theoretical guarantees and the failure mechanism induced by dither-synchronized attacks.

For every rational number $\alpha\geq 2$, we construct a 2-step nilpotent group with conjugator length function $\operatorname{CL}(n)\simeq n^\alpha$. We deduce that the nilpotent conjugator length spectrum is dense in $\{0\}\cup[2,\infty)$, even after restricting to groups of nilpotency class at most $2$. Moreover, we show that, despite being a commensurability invariant of nilpotent groups, conjugator length is not a quasi-isometry invariant: for every $m \geq 1$, the cocompact lattices in $\mathsf H(\mathbb R)^m$ realize exactly the growth types $n^2,\ldots,n^{m+1}$. Finally, for every real cubic algebraic number $\theta$, we construct a 2-step nilpotent group such that $n^3 \preceq \operatorname{CL}(n) \preceq n^{3+\varepsilon}$ for every $\varepsilon > 0$, with $n^3 \prec \operatorname{CL}(n)$ if and only if at least one real conjugate of $\theta$ has unbounded partial quotients. Consequently, determining the conjugator length function for arbitrary 2-step nilpotent groups is at least as hard as settling the bounded-partial-quotient problem for real cubic algebraic numbers with two nonreal conjugates.

Let $s(m)$ be the Conway subprime function define the binary operation on the natural numbers $x \circ y= s(x + y)$, and denote by $C_n$, $n \ge 0$, the sequence of subsets of natural numbers defined by $C_0 = \{1\}$, and $C_{n+1} = C_n \cup (C_n \circ C_n)$. We prove the conjecture by Caragiu, Vicol and Zaki that $$\lim_{n\to \infty} \frac{ | C_{n+1}| }{ | C_n |}= \frac{1+\sqrt{5}}{2}.$$ The underlying mathematical proof in this paper was constructed with some algorithmic assistance from GPT-6 Astra and its correctness has been formally verified using the Lean 4 proof assistant.

The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension $q$, every ambient dimension $D$ and every tensor order $k$, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and $D-2$, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.

We study the complexity of simple semi-conditional grammars (SSCGs) in terms of the number of their terminals and non-terminals. We show that SSCGs with three non-terminals can generate all RE languages; two non-terminals suffice for linear languages; and one suffices for unary regular languages. We also determine both upper bounds and fundamental limitations of SSCGs: while certain one-non-terminal SSCGs can generate non-context-free languages, every language generated by a one-non-terminal SSCG is in non-deterministic linear space. We also prove that there exists a regular language of alphabet size at least 3 which cannot be generated by any one-non-terminal SSCG. Finally, we prove that membership testing for SSCGs of two non-terminals is already NP-hard.

Let $\mathcal{A}^{\bf z}_{\mathfrak{g}}$ be the Gaudin algebra for the general linear Lie (super)algebra $\mathfrak{g}$ with respect to a sequence ${\bf z} \in \mathbb{C}^\ell$ of pairwise distinct complex numbers, and let $\mathbb{L}$ be any $\ell$-fold tensor product of (infinite-dimensional) unitarizable highest weight $\mathfrak{g}$-modules. Fix a singular weight $\xi$ of $\mathbb{L}$. We show that the singular weight space $\mathbb{L}^\text{sing}_\xi$ is a cyclic $\mathcal{A}^{\bf z}_{\mathfrak{g}}$-module, and that the Gaudin algebra $(\mathcal{A}^{\bf z}_{\mathfrak{g}})_{\mathbb{L}^\text{sing}_\xi}$ for $\mathbb{L}^\text{sing}_\xi$ is a Frobenius algebra. We also show that $(\mathcal{A}^{\bf z}_{\mathfrak{g}})_{\mathbb{L}^\text{sing}_\xi}$ is diagonalizable with a simple spectrum for a generic ${\bf z}$. Furthermore, we establish a super version of the Bethe ansatz for $\mathcal{A}^{\bf z}_{\mathfrak{g}}$ on $\mathbb{L}^\text{sing}_\xi$.

In this paper, a Hard Lefschetz theorem for logarithmic sheaves twisted by pseudo-effective line bundles is established, extending a theorem of Demailly, Peternell, and Schneider.

How much can a magnetic field change before a quantum spin texture loses its skyrmion charge? For a triangulated spin-$s$ texture on $N$ quantum sites, we determine the smallest change in its local moments that makes the reconstructed topological charge ill-defined, allowing unequal error bounds and fixed boundary moments. Combined with spectral perturbation theory, this geometric threshold yields explicit magnetic-field intervals that preserve the ground-state charge, scaling as $1/N$ when the gap and geometric margin have positive limits. We apply the bounds to finite spin-$1/2$ systems with exchange and Dzyaloshinskii-Moriya interactions. In a rigorously verified seven-spin example the charge changes at a single certified field, while the excitation gap and all local spin polarizations remain bounded away from zero; there the geometric margin vanishes linearly.

Distributed multiple testing asks $N$ sites, each holding p-values for its own hypotheses, to control the false discovery rate (FDR) of the discoveries made across the whole network while communicating only a small number of bits. Greedy interval aggregation (Pournaderi and Xiang, IEEE TSIPN, 2023) meets the communication budget but controls FDR only asymptotically, and its FDR can exceed the target at finite sample sizes; the same interval counts both select the rejection regions and calibrate the stopping rule, which biases the selected interval densities upward. We propose budgeted BONuS-GA: each site mixes synthetic uniform p-values into its data, the center ranks candidate p-value intervals across sites using the pooled counts, and each site's false discoveries are estimated from its own synthetic counts under a per-site share of the level. The procedure needs no knowledge of the null proportions, uses every p-value for both selection and inference, and controls $FDR\le\alpha$ at every sample size for any fixed assignment of hypotheses to sites, assuming only that the null p-values are independent uniforms, independent of the non-null p-values. It keeps the original $O(\sqrt m\log m)$ communication, $m$ being the total number of p-values, when $N=O(\sqrt m)$. We also give a sample-splitting alternative, cross-fit greedy aggregation, with finite-sample control under a random-site model, and we quantify the selection bias behind the original procedure's failure. In a monitoring-network simulation the budgeted procedure retains most of the power of its heuristic counterpart at moderate-to-large site sizes.

Motivated by Markman's work and by Kontsevich's tropical approach to disproving the Hodge conjecture, this note explains why families of Weil type abelian 2n-folds only have maximal unipotent degeneration when the discriminant is $(-1)^n$. Along the way, it explains what the discriminant is and collects various facts about Weil type abelian varieties.

In this article, we first prove that the Hecke operator $T_t$ has no eigenform in $\Sla$ with eigenvalue $-t^{k/2}$, when the characteristic of the base field is odd. Furthermore, if $\dim \Slt$ is even, we show that $T_t$ has no eigenform in $\Sla$ with eigenvalue $t^{k/2}$. As a consequence, we prove that the direct sum decomposition $\Slt=\Sold \oplus \Snew$ holds when $\dim \Slt$ is even. This proves the conjecture \cite[Conjecture 1.1(3)]{BV19a} of Bandini and Valentino for an infinite family of cusp forms. In particular, for any weight $k$, there exists at least one type $m$ (there are only two possible non-trivial values of $m$) for which the conjecture \cite[Conjecture 1.1(3)]{BV19a} is true.

We construct a big Witt theory associated with coefficient-multiplicative normalized $\sigma$-twisted Lubin-Tate formal groups. The resulting $F$-big Witt ring extends Hazewinkel's ramified $q$-typical Witt theory and recovers the usual big Witt ring in the multiplicative case. We also introduce Fleck operators defined from iterated Coleman traces and prove Lubin-Tate analogues of the Fleck-Sun-Wan congruences for both positive and negative powers. By relating the Lubin-Tate Coleman theory to the $F$-big Witt theory, we obtain a Coleman $F$-norm whose integrality yields congruences among the corresponding Fleck coefficients.

We construct a categorification of the functor of grouplike elements of a Hopf algebra in the setting of braided monoidal categories. Given a braided bistrong monoidal functor admitting a right adjoint, we define a functor assigning to each Hopf monoid a group object obtained as a subobject of its underlying object by means of finite limits. We prove that this construction is functorial and provides a right adjoint to the induced free Hopf monoid functor, thereby extending the classical correspondence between groups and grouplike elements of Hopf algebras to a categorical setting.